The completely packed O(n) loop model on the square lattice

Abstract

We present an investigation of the completely packed O(n) loop model on the square lattice by means of the transfer-matrix method and finite-size scaling. We investigate the model for a number of n values covering a wide range. This model is known to be equivalent with the q-state Potts model with q = n2, but here we also investigate the range n < 0, including rather large negative numbers. In the critical range |n| < 2, we find an energy-like scaling dimension X = 4, which is the leading one for n < 1 and the second leading one for 1 < n < 2. The point n = −2 is special, with a conformal anomaly c = −∞. For n < −2, the model is no longer critical, as evidenced e.g. by the exponentially fast convergence of the finite-size estimates of the free energy density to the infinite-system value. For |n| > 2, the system is in an ordered phase, where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence. Furthermore, we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six-vertex model.This article is part of ‘Lattice models and integrability’, a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.